I verbally accepted a job on 5/6/2015. The manager looked over my resume and said that I just need to fill out an online application. He sent an email to HR and said that I verbally accepted, so we need to get the process going. The manager said that I would start this upcoming Monday. I talked with the HR guy and he said that he still needs to call my references and that I will get a letter saying that I got accepted, but I still need to do my drug test and everything before Monday and I've got to work tomorrow and Friday at my current job.
I have no idea what to do. Is this normal?